Storing many-to-many mappings on a feed-forward neural network using fuzzy sets

Feed-forward networks are generally trained to represent functions or many-toone (m-o) mappings. In this paper however a feed-forward network with modified training algorithm is considered to represent multi-valued or one-tomany (o-m) mappings. The o-m mapping is viewed as an m-o mapping where the values corresponding to a value of the independent variable are sets. Thus the problem of representing a o-m mapping has been converted into a problem of training a network to return sets rather than vectors. The resulting o-m mapping may have variable multiplicity leading to sets of variable cardinality. The crisp sets of variable cardinality in turn are replaced by fuzzy sets of fixed cardinality by adding elements, called “do not cares” which have membership values of zero. Since the target outputs of the feedforward network are now sets of fixed cardinality and the actual output of a feedforward network is a vector the training algorithm is modified to take into account the fact that order should be removed as a constraint when the error vector is calculated. Results of simulations show that the method proposed is quite effective.